Finite Group Databasebeta

← C31×#19005 #21475764

Subgroups — C31×#19005 #21475764

Order 7936.

820 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3531C211yesmaximal, characteristic
2C31×#3239C212yesmaximal, complemented
3C31×#3139C211yesmaximal, characteristic, complemented
4C31×#1664C214yesmaximal, complemented
5C31×#1663C212yesmaximal, complemented
6C31×#1662C212yesmaximal, complemented
7C31×#1537C211yesmaximal, characteristic, complemented
8C31×#1529C212yesmaximal, complemented
9C31×#564C2211yescharacteristic
10C31×#563—22no—
11C31×#563—22no—
12C31×#563C2212yes—
13C31×#562—22no—
14C31×#562C2211yescharacteristic
15C31×#533—22no—
16C31×#533C2211yescharacteristic
17C2×C31×C4⋊Q8C2211yescharacteristic
18C31×#529—22no—
19C31×#529—22no—
20C31×#529C2212yes—
21C31×#501C2214yes—
22C31×#491C2212yes—
23C31×#491C2212yes—
24C31×#489C2214yes—
25C31×#487—22no—
26C31×#484—22no—
27C31×#481—22no—
28C31×#475—22no—
29C31×#474—22no—
30C31×#432C2212yes—
31C31×#419—22no—
32C4×M16×C31C2211yescharacteristic
33C31×#330C2212yes—
34C31×#330C2214yes—
35C31×#330C2212yes—
36C31×#329C2214yes—
37C22×Q8×C31—21no—
38C2×M16×C31—21no—
39C31×C4⋊Q8—22no—
40C31×C4⋊Q8—22no—
41C31×C4⋊Q8—22no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8D412yes—
44C31×C4⋊Q8C2312yes—
45C31×#126—22no—
46C31×#126—22no—
47C31×#126—22no—
48C31×#126—22no—
49C31×#126—22no—
50C31×#125—22no—
51C31×#125—22no—
52C31×#125—22no—
53C31×#125—22no—
54C31×#125C2311yescharacteristic
55C31×#125C2311yescharacteristic
56C31×#125C2312yes—
57C31×#124—22no—
58C31×#124—22no—
59C31×#124C2312yes—
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×#123—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8D412yes—
73C31×#121—22no—
74C31×#121—22no—
75C31×#121D412yes—
76C4×Q8×C31—22no—
77C4×Q8×C31—22no—
78C4×Q8×C31D412yes—
79C4×D4×C31—22no—
80C4×D4×C31—22no—
81C4×D4×C31D412yes—
82C31×#117—22no—
83C31×#117—22no—
84C31×#117D412yes—
85C2×C4⋊C4×C31—22no—
86C2×C42×C31—22no—
87C2×C42×C31C2311yescharacteristic
88C31×#107—24no—
89C31×#106—24no—
90C31×C4⋊C8—22no—
91C31×#103—24no—
92C31×#103—24no—
93C31×#103—24no—
94C31×#102—24no—
95C31×C22⋊C8—22no—
96C31×#97C2312yes—
97C31×#97C2314yes—
98C4×C8×C31—21no—
99C2×Q8×C31—22no—
100C2×Q8×C31—22no—
101C2×Q8×C31—22no—
102C2×Q8×C31—44no—
103C2×Q8×C31—41no—
104C2×Q8×C31C2×D412yes—
105C2×D4×C31—22no—
106C2×D4×C31—22no—
107C2×D4×C31C2×D412yes—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—41no—
112C22×C4×C31C2×D411yescharacteristic
113C22×C4×C31C2×D412yes—
114M16×C31—42no—
115C2×C8×C31—22no—
116C2×C8×C31—24no—
117C4⋊C4×C31—22no—
118C4⋊C4×C31—22no—
119C4⋊C4×C31—22no—
120C4⋊C4×C31—22no—
121C4⋊C4×C31—22no—
122C4⋊C4×C31—22no—
123C4⋊C4×C31—22no—
124C4⋊C4×C31—22no—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—22no—
127C4⋊C4×C31—22no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—22no—
130C4⋊C4×C31—24no—
131C4⋊C4×C31—42no—
132C4⋊C4×C31C2×D412yes—
133C4⋊C4×C31C2×D412yes—
134C4⋊C4×C31C2×D412yes—
135C4⋊C4×C31C2×D412yes—
136C22⋊C4×C31—22no—
137C22⋊C4×C31—22no—
138C22⋊C4×C31—22no—
139C22⋊C4×C31—22no—
140C22⋊C4×C31—22no—
141C22⋊C4×C31—22no—
142C22⋊C4×C31—22no—
143C22⋊C4×C31—22no—
144C22⋊C4×C31—22no—
145C22⋊C4×C31—22no—
146C42×C31—22no—
147C42×C31—22no—
148C42×C31—22no—
149C42×C31—22no—
150C42×C31C2×D411yescharacteristic
151C42×C31C2×D412yes—
152C42×C31C2411yescharacteristic
153#19005C3111yesmaximal, characteristic, direct product factor, Sylow
154C23×C31—22no—
155C23×C31#12011yescharacteristic
156Q8×C31—42no—
157Q8×C31—44no—
158Q8×C31—82no—
159Q8×C31—82no—
160Q8×C31—82no—
161D4×C31—42no—
162D4×C31—82no—
163D4×C31—82no—
164C2×C4×C31—22no—
165C2×C4×C31—22no—
166C2×C4×C31—22no—
167C2×C4×C31—22no—
168C2×C4×C31—22no—
169C2×C4×C31—22no—
170C2×C4×C31—42no—
171C2×C4×C31—44no—
172C2×C4×C31—42no—
173C2×C4×C31—82no—
174C2×C4×C31—82no—
175C2×C4×C31—82no—
176C2×C4×C31#12012yes—
177C2×C4×C31#12011yescharacteristic
178C2×C4×C31C22×D411yescharacteristic
179C2×C4×C31C22×D412yes—
180C8×C31—44no—
181C8×C31—42no—
182#3531C2×C3111yescharacteristic
183#3239C2×C3112yescomplemented
184#3139C2×C3111yescharacteristic, complemented
185#1664C2×C3114yescomplemented
186#1663C2×C3112yescomplemented
187#1662C2×C3112yescomplemented
188#1537C2×C3111yescharacteristic, complemented
189#1529C2×C3112yescomplemented
190C22×C31—21no—
191C22×C31—22no—
192C22×C31—42no—
193C22×C31—42no—
194C22×C31—42no—
195C22×C31C2×#12011yescharacteristic
196C4×C31—21no—
197C4×C31—22no—
198C4×C31—22no—
199C4×C31—42no—
200C4×C31—42no—
201C4×C31—42no—
202C4×C31—42no—
203C4×C31—42no—
204C4×C31—82no—
205C4×C31#45412yes—
206#564C22×C3111yescharacteristic
207#563—22no—
208#563—22no—
209#563C22×C3112yes—
210#562—22no—
211#562C22×C3111yescharacteristic
212#533—22no—
213#533C22×C3111yescharacteristic
214C2×C4⋊Q8C22×C3111yescharacteristic
215#529—22no—
216#529—22no—
217#529C22×C3112yes—
218#501C22×C3114yes—
219#491C22×C3112yes—
220#491C22×C3112yes—
221#489C22×C3114yes—
222#487—22no—
223#484—22no—
224#481—22no—
225#475—22no—
226#474—22no—
227#432C22×C3112yes—
228#419—22no—
229C4×M16C22×C3111yescharacteristic
230#330C22×C3112yes—
231#330C22×C3112yes—
232#330C22×C3114yes—
233#329C22×C3114yes—
234C2×C31—41no—
235C2×C31—82no—
236C2×C31C2×#45411yescharacteristic
237C2×C31#302212yes—
238C22×Q8—21no—
239C2×M16—21no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—21no—
244C4⋊Q8D4×C3112yes—
245C4⋊Q8C23×C3112yes—
246#126—22no—
247#126—22no—
248#126—22no—
249#126—22no—
250#126—22no—
251#125—22no—
252#125—22no—
253#125—22no—
254#125—22no—
255#125C23×C3112yes—
256#125C23×C3111yescharacteristic
257#125C23×C3111yescharacteristic
258#124—22no—
259#124—22no—
260#124C23×C3112yes—
261#123—22no—
262#123—22no—
263#123—22no—
264#123—22no—
265#123—22no—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8D4×C3112yes—
274#121—22no—
275#121—22no—
276#121D4×C3112yes—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8D4×C3112yes—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4D4×C3112yes—
283#117—22no—
284#117—22no—
285#117D4×C3112yes—
286C2×C4⋊C4—22no—
287C2×C42—22no—
288C2×C42C23×C3111yescharacteristic
289#107—24no—
290#106—24no—
291C4⋊C8—22no—
292#103—24no—
293#103—24no—
294#103—24no—
295#102—24no—
296C22⋊C8—22no—
297#97C23×C3112yes—
298#97C23×C3114yes—
299C4×C8—21no—
300C31#1900511yescharacteristic, direct product factor, Sylow
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—44no—
305C2×Q8—41no—
306C2×Q8C2×D4×C3112yes—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4C2×D4×C3112yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4C2×D4×C3111yescharacteristic
315C22×C4C2×D4×C3112yes—
316M16—42no—
317C2×C8—22no—
318C2×C8—24no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—42no—
334C4⋊C4C2×D4×C3112yes—
335C4⋊C4C2×D4×C3112yes—
336C4⋊C4C2×D4×C3112yes—
337C4⋊C4C2×D4×C3112yes—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42C2×D4×C3111yescharacteristic
353C42C2×D4×C3112yes—
354C42C24×C3111yescharacteristic
355C23—22no—
356C23C31×#12011yescharacteristic
357Q8—42no—
358Q8—44no—
359Q8—82no—
360Q8—82no—
361Q8—82no—
362D4—42no—
363D4—82no—
364D4—82no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4C31×#12012yes—
378C2×C4C31×#12011yescharacteristic
379C2×C4C22×D4×C3111yescharacteristic
380C2×C4C22×D4×C3112yes—
381C8—44no—
382C8—42no—
383C22—21no—
384C22—22no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22C2×C31×#12011yescharacteristic
389C4—21no—
390C4—22no—
391C4—22no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—82no—
398C4C31×#45412yes—
399C2—41no—
400C2—82no—
401C2C2×C31×#45411yescharacteristic
402C2C31×#302212yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3531C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C31×#3239C2semidirect
3. C31×#3139C2semidirect
4. C31×#1664C2semidirect
5. C31×#1663C2semidirect
6. C31×#1662C2semidirect
7. C31×#1537C2semidirect
8. C31×#1529C2semidirect
153. #19005C31direct (N × H)
183. #3239C2×C31semidirect
184. #3139C2×C31semidirect
185. #1664C2×C31semidirect
186. #1663C2×C31semidirect
187. #1662C2×C31semidirect
188. #1537C2×C31semidirect
189. #1529C2×C31semidirect
300. C31#19005direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C31×#35312
C31×#32392
C31×#31392
C31×#16642
C31×#16632
C31×#16622
C31×#15372
C31×#15292
#1900531